Let F be a collection of disjoint intervals of real numbers. This means that if I and J are two distinct intervals in F, then the intersection of I and J is empty?. Prove that the set F is countable.
Between any two real numbers there is a rational number.
Properly defined a closed interval looks like .
Thus in each closed interval in the collection there is a rational number.
Because the set of rationals is countable the collection is countable.